Presentation on the topic octahedron. Properties of the dodecahedron and interesting facts
Definition: A convex polyhedron is called
correct if all its faces are
equal regular polygons and in
at each of its vertices the same thing converges
same number of ribs. Correct
There are only five polyhedra: tetrahedron,
hexahedron, octahedron, dodecahedron, icosahedron.
Tetrahedron
Octahedron
A tetrahedron is the simplest polyhedron with faces
which are four triangles. U
tetrahedron has 4 faces, 4 vertices and 6 edges. Tetrahedron, y
of which all faces are equilateral
triangles are called
correct. The right one
tetrahedron all dihedral angles at edges and
all trihedral angles at the vertices are equal.
Octahedron - has 8 triangular faces, 12 edges, 6
vertices, 4 edges converge at each vertex.
Examples of regular polyhedra:
IcosahedronCube
Icosahedron - regular convex
polyhedron, twenty-hedron. Each of 20
faces represents
equilateral triangle. The number of edges is
30, number of vertices - 12. The icosahedron has
59 star shapes.
A cube is a regular polyhedron, each face
which is a square. Vershin -
8, Edges - 12, Faces - 6.
Examples of regular polyhedra:
DodecahedronDodecahedron - composed of
twelve correct
pentagons that are his
edges.
Each vertex of the dodecahedron
is the top of the three right
pentagons. Thus,
dodecahedron has 12 faces
(pentagonal), 30 edges and 20
vertices (3 edges converge at each).
Characteristics and formulas:
Elements of symmetry of a regular tetrahedron:
A regular tetrahedron has no center
symmetry. But it has three axes
symmetry and six planes
symmetry.
Elements of symmetry of a regular octahedron:
A regular octahedron has a centersymmetry - the point of intersection of its axes
symmetry. Three of 9 planes
the symmetries of the tetrahedron pass through
every 4 vertices of the octahedron lying in
one plane. Six planes
symmetries pass through two vertices,
not belonging to the same face, and
the middle of the opposite ribs.
Elements of symmetry of a regular icosahedron:
A regular icosahedron has 15 axessymmetries, each of which passes
through the middles of opposite
parallel ribs. Intersection point
of all axes of symmetry of the icosahedron is
its center of symmetry. Planes
symmetry also 15. Planes
symmetries pass through four
vertices lying in the same plane, and
midpoints of opposite parallels
ribs
Cube symmetry elements:
The cube has one center of symmetry -the point of intersection of its diagonals, also
9 axes pass through the center of symmetry
symmetry. Planes of symmetry of a cube
also 9 and they pass either through
opposite ribs.
Elements of symmetry of a regular dodecahedron:
A regular dodecahedron has a centersymmetry and 15 axes of symmetry. Each
of the axes passes through the midpoints
opposite parallel ribs.
The dodecahedron has 15 planes
symmetry. Any of the planes
symmetry runs in every face
through the top and middle
opposite rib.
All information taken from:
http://licey102.k26.ru/http://math4school.ru
wikipedia.org
Textbook for grades 10-11 on geometry
One of the oldest mentions of regular polyhedra is in Plato's (BC) treatise Timaus. Therefore, regular polyhedra are also called Platonic solids (although they were known long before Plato). Each of the regular polyhedra, and there are five in total. Plato associated with four “earthly” elements: earth (cube), water (icosahedron), fire (tetrahedron), air (octahedron), as well as with the “unearthly” element - sky (dodecahedron).
A regular polyhedron, or Platonic solid, is a convex polyhedron with the greatest possible symmetry. A polyhedron is called regular if: it is convex, all its faces are equal regular polygons and converge at each of its vertices same number all its dihedral angles are equal
Let us note an interesting fact related to the hexahedron (cube) and octahedron. A cube has 6 faces, 12 edges and 8 vertices, and an octahedron has 8 faces, 12 edges and 6 vertices. That is, the number of faces of one polyhedron is equal to the number of vertices of another and vice versa. As they say, the cube and the hexahedron are dual to each other. This is also manifested in the fact that if you take a cube and build a polyhedron with vertices at the centers of its faces, then, as you can easily see, you get an octahedron. The reverse is also true - the centers of the octahedron faces serve as the vertices of the cube. This is the duality of the octahedron and the cube (fig). It is easy to figure out that if we take the centers of the faces of a regular tetrahedron, we will again get a regular tetrahedron (fig). Thus, the tetrahedron is dual to itself.
The famous mathematician and astronomer Kepler built a model solar system as a series of sequentially inscribed and described regular polyhedra and spheres. What order of arrangement of planets (in accordance with the “requirements” of regular polyhedra) did Kepler obtain? A cube was inscribed in the sphere of the orbit of Saturn, and the sphere of the orbit of Jupiter was inscribed in it; the tetrahedron fits into this sphere, and the sphere of the orbit of Mars fits into it; further: dodecahedron - sphere of the Earth's orbit - icosahedron - sphere of the orbit of Venus - octahedron - sphere of the orbit of Mercury.
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Slide captions:
A polyhedron is a body whose surface consists of a finite number of flat polygons.
Regular polyhedra
How many regular polyhedra are there? - How are they determined, what properties do they have? -Where are they found, do they have practical applications?
A convex polyhedron is called regular if all its faces are equal regular polygons and the same number of edges converge at each of its vertices.
"hedra" - face "tetra" - four hexes - six "octa" - eight "dodeca" - twelve "icos" - twenty The names of these polyhedra came from Ancient Greece and they indicate the number of faces.
Name of regular polyhedron Type of face Number of vertices of edges of faces of faces converging at one vertex Tetrahedron Regular triangle 4 6 4 3 Octahedron Regular triangle 6 12 8 4 Icosahedron Regular triangle 12 30 20 5 Cube (hexahedron) Square 8 12 6 3 Dodecahedron Regular pentagon 20 30 12 3 Data on regular polyhedra
Question (problem): How many regular polyhedra are there? How to set their number?
α n = (180 °(n -2)): n At each vertex of the polyhedron there are at least three plane angles, and their sum must be less than 360 °. Shape of faces Number of faces at one vertex Sum of plane angles at the vertex of a polyhedron Conclusion about the existence of a polyhedron α = 3 α = 4 α = 5 α = 6 α = 3 α = 4 α = 3 α = 4 α = 3
L. Carroll
Great mathematicians of antiquity Archimedes Euclid Pythagoras
The ancient Greek scientist Plato described in detail the properties of regular polyhedra. That is why regular polyhedra are called Platonic solids
tetrahedron - fire cube - earth octahedron - air icosahedron - water dodecahedron - universe
Polyhedra in space and earth sciences
Johannes Kepler (1571-1630) – German astronomer and mathematician. One of the founders of modern astronomy - discovered the laws of planetary motion (Kepler's laws)
Kepler Cup Cosmic
"Ecosahedron - dodecahedral structure of the Earth"
Polyhedra in art and architecture
Albrecht Durer (1471-1528) "Melancholy"
Salvador Dali "The Last Supper"
Modern architectural structures in the form of polyhedra
Alexandrian lighthouse
Brick polyhedron by a Swiss architect
Modern building in England
Polyhedra in nature FEODARIA
Pyrite (sulfur pyrite) Monocrystal of potassium alum Crystals of red copper ore NATURAL CRYSTALS
Table salt consists of cube-shaped crystals. The mineral sylvite also has a cube-shaped crystal lattice. Water molecules are shaped like a tetrahedron. The mineral cuprite forms crystals in the shape of octahedrons. Pyrite crystals have the shape of a dodecahedron
Diamond In the form of an octahedron, diamond, sodium chloride, fluorite, olivine and other substances crystallize.
Historically, the first cut form that appeared in the 14th century was the octahedron. Diamond Shah Diamond weight 88.7 carats
Task British Queen gave instructions to cut along the edges of the diamond with gold thread. But the cutting was not done, since the jeweler was unable to calculate maximum length gold thread, but the diamond itself was not shown to him. The jeweler was informed of the following data: number of vertices B = 54, number of faces D = 48, length of the largest edge L = 4 mm. Find the maximum length of the golden thread.
Regular polyhedron Number of Faces Vertices Edges Tetrahedron 4 4 6 Cube 6 8 12 Octahedron 8 6 12 Dodecahedron 12 20 30 Icosahedron 20 12 30 Research"Euler's Formula"
Euler's theorem. For any convex polyhedron B + G - 2 = P where B is the number of vertices, G is the number of faces, P is the number of edges of this polyhedron.
PHYSICAL MINUTE!
Problem Find the angle between two edges of a regular octahedron that have a common vertex but do not belong to the same face.
Problem Find the height of a regular tetrahedron with an edge of 12 cm.
The crystal has the shape of an octahedron, consisting of two regular pyramids with a common base, the edge of the base of the pyramid is 6 cm. The height of the octahedron is 8 cm. Find the lateral surface area of the crystal
Surface area Tetrahedron Icosahedron Dodecahedron Hexahedron Octahedron
Homework assignment: mnogogranniki.ru Using developments, make models of the 1st regular polyhedron with a side of 15 cm, 1st semiregular polyhedron
Thanks for the work!
A polyhedron is a surface composed of polygons that bound a geometric body. Polyhedra are convex and non-convex polygons. A polyhedron is called convex if it is located on one side of the plane of each polygon on its surface
Octahedron The octahedron (Greek οκτάεδρον, from Greek οκτώ, “eight” and Greek έδρα “base”) is one of the five convex regular polyhedra, the so-called Platonic solids. regular Platonic polyhedra The octahedron has 8 triangular faces, 12 edges, 6 vertices, and 4 edges converge at each vertex.
Icosahedron Icosahedron (from the Greek εικοσάς twenty; -εδρον face, face, base) is a regular convex polyhedron, twenty-hedron, one of the Platonic solids. Each of the 20 faces is an equilateral triangle. The number of edges is 30, the number of vertices is 12. The icosahedron has 59 stellated shapes. Greek Platonic solids triangle stellated shapes
Dodecahedron Dodecahedron (from the Greek δώδεκα twelve and εδρον face), dodecahedron is a regular polyhedron composed of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. Greek regular polyhedron of regular pentagons vertex Thus, the dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each). The sum of the plane angles at each of the 20 vertices is equal to 324°.angles
Slide 1
Slide 2
SYMMETRY IN SPACE “Symmetry is the idea through which man tried to comprehend and create order, beauty and perfection” (G. Weil) Symmetry (“proportionality”) is correspondence, immutability (invariance), manifested during any transformations. For example, the spherical symmetry of a body means that the appearance of the body will not change if it is rotated in space at arbitrary angles, keeping one point in place. "Vitruvian Man" by Lenardo Da Vinci (1490, Venice)
Slide 3
SYMMETRY IN SPACE Points A and A1 are called symmetrical relative to point O (center of symmetry) if O is the middle of segment AA1. Point O is considered symmetrical to itself. A A1
Slide 4
SYMMETRY IN SPACE Points A and A1 are called symmetrical with respect to a straight line (axis of symmetry) if the straight line passes through the middle of the segment AA1 and is perpendicular to this segment. Each point of a line a is considered symmetrical to itself. A1
Slide 5
SYMMETRY IN SPACE Points A and A1 are called symmetrical relative to a plane (plane of symmetry) if this plane passes through the middle of the segment AA1 and is perpendicular to this segment. Each point of the plane is considered symmetrical to itself
Slide 6
SYMMETRY IN SPACE A point (straight line, plane) is called the center (axis, plane) of symmetry of a figure if each point of the figure is symmetrical relative to it to some point of the same figure. If a figure has a center (axis, plane) of symmetry, then it is said to have central (axial, mirror) symmetry
Slide 7
EXAMPLES OF SYMMETRY OF PLATE FIGURES A parallelogram has only central symmetry. Its center of symmetry is the point of intersection of the diagonals. An equilateral trapezoid has only axial symmetry. Its axis of symmetry is a perpendicular drawn through the midpoints of the bases of the trapezoid. A rhombus has both central and axial symmetry. Its axis of symmetry is any of its diagonals; center of symmetry - the point of their intersection
Slide 8
REGULAR POLYHEDRONS - 5 PLATONIAN SOLIDS The inhabitants of even the most distant galaxy cannot play dice, which have the shape of a regular convex polyhedron unknown to us. M. Gardner A convex polyhedron is called regular if all its faces are equal regular polygons and the same number of edges converge at each of its vertices. Also, all edges of a regular polygon are equal, as are all dihedral angles containing two faces with a common edge. A regular polyhedron whose faces are n-gons for n > or = 6 does not exist!
Slide 9
REGULAR TETRAHEDER Composed of four equilateral triangles. Each of its vertices is the vertex of three triangles. The sum of the plane angles at each vertex is exactly 180°. Elements of symmetry: The tetrahedron does not have a center of symmetry, but has 3 axes of symmetry and 6 planes of symmetry. S full Volume Height of Vertices – 4 Faces – 6 Edges – 4
Slide 10
CUBE Made up of six squares. Each vertex of the cube is the vertex of three squares. The sum of the plane angles at each vertex is exactly 270°. 6 faces, 8 vertices and 12 edges Elements of symmetry: The cube has a center of symmetry - the center of the cube, 9 axes and planes of symmetry R description. env. S full r in. okr
Slide 11
REGULAR OCTAHEDRON Composed of eight equilateral triangles. Each vertex of the octahedron is the vertex of four triangles. The sum of the plane angles at each vertex is 240°. Elements of symmetry: The octahedron has a center of symmetry - the center of the octahedron, 9 axes of symmetry and 9 planes of symmetry 8 faces 6 vertices 12 edges
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